The Moon accreted meteoritic material towards the end of Solar System formation. Quantification of this late accretion requires an estimation of the abundance of highly siderophile, or iron-loving, elements in the lunar mantle. As lunar mantle samples are not available, estimates are derived from lunar basalt compositions, but the melting phase relations needed to derive the mantle composition are poorly constrained. Here we present sulfur solubility measurements from laboratory experiments, combined with thermodynamic calculations, which show that the lunar basalt source is likely to be saturated in a sulfur-poor, iron-rich sulfide melt that concentrates some highly siderophile elements more than others. We found that the observed range in the ratios of highly siderophile elements in primitive lunar basalts is much smaller than expected from residual sulfide control alone. Instead, the elemental ratios are consistent with mixing between primary sulfide-saturated melts and minute (<1%) amounts of lunar regolith that contain impact debris. Although the composition of some samples suggests a highly depleted lunar mantle, the exact level of depletion is unclear, because mixing trajectories overlap at the inferred level of regolith contamination. We conclude that the composition of the lunar mantle is veiled by regolith contamination of the lunar basalts. If so, highly siderophile element abundances in lunar mantle-derived materials cannot be used to determine the mass of material accreted late onto the Moon. The abundance of iron-loving elements in Moon rocks cannot indicate the amount of late accretion onto the Moon, according to experiments and thermodynamic calculations constraining the behaviour of these elements under lunar mantle conditions.